Suppose you just purchased a used car, and the number of miles on the odometer can be represented by the equation
, where
is the number of miles on the odometer, and
is the number of miles you have driven it. Could you convert this equation to function notation? How many miles will be on the odometer if you drive the car 700 miles? In this Concept, you'll learn how to convert equations such as this one to function notation and how to input a value into a function in order to get an output value.

Guidance

So far, the term
function
has been used to describe many of the equations we have been graphing. The concept of a function is extremely important in mathematics. Not all equations are functions. To be a function, for each value of
there is one and only one value for
.

Definition:
A
function
is a relationship between two variables such that the input value has ONLY one unique output value.

Recall from a previous Concept that a function rule replaces the variable
with its function name, usually
. Remember that these parentheses do not mean multiplication. They separate the function name from the independent variable,
.

is read “the function
of
” or simply “
of
.”

If the function looks like this:
, it would be read
of
equals 3 times
minus 1.

Using Function Notation

Function notation allows you to easily see the input value for the independent variable inside the parentheses.

Example A

Consider the function
.

Evaluate
.

Solution:
The value inside the parentheses is the value of the variable
. Use the Substitution Property to evaluate the function for
.

To use function notation, the equation must be written in terms of
. This means that the
variable must be isolated on one side of the equal sign.

Example B

Rewrite
using function notation.

Solution:
The goal is to rearrange this equation so the equation looks like
. Then replace
with
.

Functions as Machines

You can think of a function as a machine. You start with an input (some value), the machine performs the operations (it does the work), and your output is the answer. For example,
takes
some number
,
, multiplies it by 3 and adds 2. As a machine, it would look like this:

When you use the function machine to evaluate
, the solution is
.

Example C

A function is defined as
. Determine the following:

a)

b)

Solution:

a) Substitute
into the function
.

b) Substitute
into the function
.

Video Review

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Guided Practice

Rewrite the equation
in function notation where
, and then evaluate
, and
.

Explore More

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.
CK-12 Basic Algebra: Linear Function Graphs
(11:49)